Bounded continuously distributed delays in dynamic oligopolies

Chaos, Solitons and Fractals 18 (2003) 977–993

www.elsevier.com/locate/chaos

Bounded continuously distributed delaysin dynamic oligopolies

Carl Chiarella a,*, Ferenc Szidarovszky b

a School of Finance and Economics, University of Technology, Sydney, P.O. Box 123, Broadway, NSW 2007, Australiab Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ 85721-0020, USA

Accepted 27 February 2003

Abstract

This paper introduces a way of modelling bounded continuously distributed time lags in dynamic economic models.

Past data are averaged only over a bounded interval thus avoiding the use of very old (‘‘stale’’) economic data and

hence making dynamic economic models more realistic. Dynamic oligopolies are formulated and then examined under

this new type of information lag. Stability analysis is presented and the possibility of the birth of limit cycles is ex-

amined. Some special cases are considered, the computational analysis of which illustrates the theoretical findings.

� 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction

Dynamic oligopoly models have been examined by many researchers during the last three decades. The most im-

portant references and results on single-product oligopolies are summarised in [7] and their extensions to the multi-

product case are discussed in [8]. Most of the earlier models were based on the special assumption that at each time

period all relevant information is instantaneously available. However in economic reality there is always a time lag in

obtaining and implementing information about the rivals� output as well as about the firms� own output. Time delays

can be modelled either by assuming fixed time lags or continuously distributed time lags. Russel et al. [10] used fixed

time lags and their dynamic model was described by a differential-difference equation. The major difficulty in using this

idea arises from the fact that time delays are not known exactly. Continuously distributed time lags are more realistic.

They have been earlier introduced and used in mathematical biology (see for example [4]), and their first economic

application was presented by Invernizzi and Medio [6]. Based on their ideas Chiarella and Khomin [2] and Chiarella

and Szidarovszky [3] have examined dynamic oligopolies with continuously distributed time lags and examined the

asymptotic behaviour of the equilibrium. Continuously distributed time lags are based on the weighted average of all

past data from time zero up to the current time period t. For larger values of t this assumption seems unrealistic since

very early, irrelevant data are also used in calculating the current average. In addition, dynamic economic models with

continuously distributed time lags right back to the beginning of the process generally tend to be highly stable. The aim

of this paper is to introduce continuously distributed lags over a finite time interval. Dynamic oligopoly models will be

formulated with these bounded continuously distributed lags and their stability properties, particularly in relation to the

possible birth of limit cycles, will be studied and compared with the infinitely continuously distributed lag case con-

sidered by Chiarella and Szidarovszky [3].

* Corresponding author. Tel.: +61-2-9514-7719; fax: +61-2-9514-7711.

E-mail addresses: [email protected] (C. Chiarella), [email protected] (F. Szidarovszky).

0960-0779/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0960-0779(03)00067-5

mail to: [email protected]

978 C. Chiarella, F. Szidarovszky / Chaos, Solitons and Fractals 18 (2003) 977–993

This paper develops as follows. In Section 2 bounded continuously distributed lags will be introduced, and in Section

3 will be applied to the corresponding dynamic oligopoly models. We will also show that the resulting integro-differ-

ential equations are equivalent to a certain system of ordinary differential-difference equations. In Section 4 stability

analysis on the basis of local linearization will be undertaken, and the Hopf bifurcation theorem will be applied

in Section 5 to examine the possibility of the birth of limit cycles. Some important special cases will be discussed in

Section 6, and in Sections 7 and 8 computational results will illustrate the theoretical findings. Section 9 will conclude

the paper.

2. Bounded continuously distributed lags

Select a positive constant d > 0, and assume that the delayed information on any function xðtÞ is given by a weighted

average of the past data xðsÞ for s 2 ½t � d; t�. The weighting function of infinitely continuously distributed lags used in

[2,3,6] cannot be used here, since its integral equals 1 on the interval ½0;1�, rather than on the new interval ½t � d; t�.However we may still use basically the same weighting function if an appropriate normalising factor is introduced.

Hence in this paper we will use weighting functions of the form

wðt � s; T ; 0; dÞ ¼ 1

Cð0; T ; dÞ 1

Te�

ðt�sÞT for m ¼ 0 ð1Þ

and

wðt � s; T ;m; dÞ ¼ 1

Cðm; T ; dÞ 1

m!mT

� �mþ1

ðt � sÞme�mðt�sÞ

T for m ¼ 1; 2; . . . ð2Þ

Here Cðm; T ; dÞ is the normalising factor and we note that functions (1) and (2) are the 1=Cðm; T ; dÞ-multiples of the

corresponding weighting functions used by Invernizzi and Medio [6], Chiarella and Khomin [2] and Chiarella and

Szidarovszky [3] in modelling continuously distributed time lags. The weighting functions have the following properties:

ii(i) for m ¼ 0, weights are exponentially declining with the most weight given to the most current data;

i(ii) for mP 1, zero weight is assigned to the most recent output, rising to maximum at t � s ¼ T , and declining expo-

nentially thereafter;

(iii) as m increases, the function becomes more peaked around t � s ¼ T . For sufficiently large values of m the function

may for all practical purposes be regarded as very close to the Dirac delta function centered at t � s ¼ T ;(iv) as T ! 0, the function tends to the Dirac delta function.

First we establish the functional form of the normalising factors.

Lemma 1. The normalising factors are given by

Cð0; T ; dÞ ¼ 1� e�dT for m ¼ 0

and for mP 1, by

Cðm; T ; dÞ ¼ 1� e�mdT

Xmk¼0

mdT

� �kk!

:

Proof. The normalising factors result from the requirement thatR tt�d wðt � s; T ;mÞds ¼ 1 for all m.

Notice first that

Cð0; T ; dÞ ¼Z t

t�d

1

Te�

t�sT ds ¼

Z 0

d� 1

Te�

ud du ¼ 1� e�

dT ;

when we introduce the new variable u ¼ t � s.In the general case notice first that

Cðm; T ; dÞ ¼Z t

t�d

1

m!mT

� �mþ1


T ds ¼Z d

0

1

m!mT

� �mþ1

ume�muT du

0.00 0.05 0.10 0.15 0.20 0.25t-s

0

5

10

15

20

25

30

35

40

w

m = 0 2 4 6 8 10

δ = 0.25T = 1

0

1

2

3

4

5

6

w

0.00 0.15 0.30 0.45 0.60 0.75t-s

T = 1 δ = 0.75m = 0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

w

0.25 0.50 0.75 1.00 1.25t-s

0.0

0.5

1.0

1.5

w

0.00 0.50 1.00 1.50t-s

m = 0 2 4 6 8 10

T = 1 δ = 1.5T = 1 δ = 1.25

m = 0 2 4 6 8 10

0.00

Fig. 1. The bounded continuously distributed lag weighting function for various values of d and m.

C. Chiarella, F. Szidarovszky / Chaos, Solitons and Fractals 18 (2003) 977–993 979

and by introducing the integrals

1 Sin

the fam

Ck ¼Z d

0

1

k!mT

� �kþ1

uke�muT du ðk ¼ 0; 1; 2; . . . ;mÞ

we see that Cðm; T ; dÞ ¼ Cm.

Simple integration shows that

C0 ¼Z d

0

mTe�

muT du ¼ 1� e�

mdT

and by using integration by parts we have for all k P 1,

Ck ¼1

k!mT

� �kþ1

uke�

muT

� mT

� d

0

�Z d

0

1

k!mT

� �kþ1

kuk�1 e�mu

T

� mT

du ¼ � 1

k!mT

� �kdke�

mdT þ Ck�1:

The assertion follows by adding this equality for k ¼ 1; 2; . . . ;m. �

We illustrate the bounded continuously distributed lag weighting function wðt � s; T ;mÞ in Fig. 1 for m varying from

0 to 10 at various values of d. We note that the maximum (for m > 0) of wðt � s; T ;mÞ occurs at T and here we have

taken T ¼ 1. 1 So for values of d < 1, the past values are given an increasing weight as one moves from t to t � d; this ismerely a reflection of the fact that over such a (relatively) short interval the most recent observations may be regarded

ce the time scale can always be adjusted to make T ¼ 1, Figs. 1 and 2 can be regarded as giving a feel for the general nature of

ily of weighting functions.

0.0

0.5

1.0

1.5

w

0 1 2 3 4 5t-s

0.0

0.5

1.0

1.5

w

0 1 2 3 4 5t-s

m 10

8

6

4

2

0

T = 1 δ = ∞T = 1 δ = 5 m 10

8

6

4

2 0

Fig. 2. Comparing large d with the infinitely continuously distributed lag weighting function.


as less reliable or less certain by the economic agents. As d increases past T , the weighting function then drops off. In

this case agents give less weight to most recent information for reasons previously stated; they give maximum weight to

information around lag T ; information beyond lag T is given decreasing weight being regarded as less relevant to

current circumstances. Thus Fig. 1 illustrates clearly how the bounded continuously distributed lag weighting function

allows economic agents to concentrate on those parts of past information that are considered most relevant and weight

that information as they feel appropriate by adjusting the parameter m.It is worth pointing out that for all mP 0

limd!1

Cðm; T ; dÞ ¼ 1

so that in this limit the weighting functions (1) and (2) reduce to the infinitely continuously distributed lag case.

In Fig. 2 we compare the cases d ¼ 5 and d ¼ 1, still for T ¼ 1. There is essentially no difference between the two

weighting functions. This observation will be useful later when we seek to illustrate the difference in dynamic behaviour

between the bounded and infinitely continuously distributed lag situations.

3. Dynamic oligopolies

Consider an n-firm single-product oligopoly without product differentiation in which xiðtÞ is the output of firm i attime period t, QiðtÞ ¼

Pj6¼i xjðtÞ is the output of the rest of the industry, and giðQiðtÞÞ is the best response of firm i. The

equilibrium of the oligopoly is a constant vector x ¼ ðx 1; . . . ; x nÞ such that for all i ¼ 1; 2; . . . ; n,

x i ¼ giXj6¼i

x j

!:

In the local stability analysis to be considered in subsequent sections, the slopes of the best response functions at the

equilibrium, namely

ci ¼ g0iðQ i Þ

play an important role. In this paper we only consider oligopolies for which the best response function is downward

sloping, so ci < 0.

Assume that at each time period each firm adjusts its output in the direction of the expected best response then the

following dynamic equations are obtained:

_xxiðtÞ ¼ kiðgiðQei ðtÞÞ � xei ðtÞÞ ði ¼ 1; 2; . . . ; nÞ; ð3Þ

where ki > 0 is a given speed of adjustment for each firm. Here Qei ðtÞ and xei ðtÞ are respectively firm i�s expectation of rest

of industry output and its own output at time t. Both Qei ðtÞ and xei ðtÞ are based on delayed information, and the delays

are bounded continuously distributed as described in Section 2. Thus


Qei ðtÞ ¼

Z t

t�di

wðt � s; Ti;mi; diÞQiðsÞds ð4Þ

and

xei ðtÞ ¼Z t

t�Di

wðt � s; Si; li;DiÞxiðsÞds: ð5Þ

We use di and Di to respectively denote for firm i the length of the interval over which past rest of industry output

and its own output enter its decision process. As Fig. 1 indicates, the values of Ti and Si show that among all past data

being considered, Qiðt � TiÞ and xiðt � SiÞ have the largest weights for mi P 1 and li P 1. The values of mi and li indicatehow peaked the weighting functions are around Ti and Si, respectively.

Substituting these equations into (3) a non-linear integro-differential equation system is obtained, the local stability

of which is the subject of the next section.

It is clear that a constant vector x ¼ ðx 1; . . . ; x nÞ is an equilibrium of the n-firm oligopoly if and only if it is the

steady-state of the dynamic system (3).

We will next show that all techniques for analysing the asymptotic properties of ordinary differential-difference

equations (expounded by Bellman and Cooke [1]) can be applied by verifying that equation (3) are equivalent to a

system of non-linear ordinary differential-difference equations.

Lemma 2. The system of integro-differential equation (3) is equivalent to the system of 3nþPn

i¼1ðmi þ liÞ ordinary dif-ferential-difference equations (8), (9), (15), (16) and (20) below.

Proof. Assume first that mi P 1. For k ¼ 1; 2; . . . ;mi introduce functions

qðkÞi;miðtÞ ¼

Z t

t�di

1

k!mi

Ti

�kþ1

ðt � sÞke�miðt�sÞ

Ti QiðsÞds; ð6Þ

then the definition of the weighting functions w implies that

Qei ðtÞ ¼

1

Cðmi; Ti; diÞqðmiÞi;mi

ðtÞ: ð7Þ

By simple differentiation we see that

_qqð0Þi;miðtÞ ¼ mi

Ti

Xj 6¼i

xjðtÞ"

� qð0Þi;miðtÞ#� mi

Tie�midi

Ti

Xj 6¼i

xjðt � diÞ ð8Þ

and for k P 1

_qqðkÞi;miðtÞ ¼ mi

Tiqðk�1Þi;mi

ðtÞh

� qðkÞi;miðtÞi� 1

k!mi

Ti

�kþ1

dki e

�midiTi

Xj 6¼i

xjðt � diÞ: ð9Þ

In the case of mi ¼ 0, we have

Qei ðtÞ ¼

1

Cð0; Ti; diÞqð0Þi;0 ðtÞ ð10Þ

with

qð0Þi;0 ðtÞ ¼Z t

t�di

1

Tie�ðt�sÞ

Ti QiðsÞds: ð11Þ

Then simple differentiation shows that

_qqð0Þi;0 ðtÞ ¼1

Ti

Xj 6¼i

xjðtÞ"

� qð0Þi;0 ðtÞ#� 1

Tie�di

Ti

Xj 6¼i

xjðt � diÞ: ð12Þ


Assume now that li P 1. Similarly to the previous derivation introduce functions

2 Th

xðkÞi;li ðtÞ ¼Z t

t�Di

1

k!liSi

�kþ1

ðt � sÞke�liðt�sÞ

Si xiðsÞds; ð13Þ

then obviously

xei ðtÞ ¼1

Cðli;; Si;DiÞxðliÞi;li ðtÞ ð14Þ

and similar equations hold for _xxðkÞi;li ðtÞ. Thus

_xxð0Þi;li ðtÞ ¼liSi

xiðtÞh

� xð0Þi;li ðtÞi� liSie�liDi

Si xiðt � DiÞ ð15Þ

and for kP 1,

_xxðkÞi;li ðtÞ ¼liSi

xðk�1Þi;li ðtÞ

h� xðkÞi;li ðtÞ

i� 1

k!liSi

�kþ1

Dki e

�liDiSi xiðt � DiÞ: ð16Þ

In the case of li ¼ 0, we have

xei ðtÞ ¼1

Cð0; Si;DiÞxð0Þi;0 ðtÞ ð17Þ

with

xð0Þi;0 ðtÞ ¼Z t

t�Di

1

Sie�ðt�sÞ

Si xiðsÞds: ð18Þ

By simple differentiation

_xxð0Þi;0 ðtÞ ¼1

SixiðtÞh

� xð0Þi;0 ðtÞi� 1

Sie�Di

Si xiðt � DiÞ: ð19Þ

By using relations (7) and (14) we can rewrite the dynamic equation (3) in terms of the new variables as

_xxiðtÞ ¼ ki gi1

Cðmi; Ti; diÞqðmiÞi;mi

ðtÞ ��

� 1

Cðli; Si;DiÞxðliÞi;li ðtÞ

: ð20Þ

Then Eqs. (8), (9), (15), (16) and (20) give a system of ordinary differential-difference equations 2 for the unknown

functions qðkÞi;miðtÞ ðk ¼ 0; 1; . . . ;miÞ, xðkÞi;li ðtÞ ðk ¼ 0; 2; . . . ; liÞ, and xiðtÞ for i ¼ 1; 2; . . . ; n with Qe

i ðtÞ and xei ðtÞ calculated

from (7) and (14) respectively. Notice that the number of equations equals the number of unknowns, namely

Xni¼1

ððmi þ 1Þ þ ðli þ 1Þ þ 1Þ ¼ 3nþXni¼1

ðmi þ liÞ:

The initial conditions are selected as follows:

qðkÞi ð0Þ ¼ 0 ðfor all kÞ;xðkÞi ð0Þ ¼ 0 ðfor all kÞ;xiðsÞ ¼ close to equilibrium ð�e6 s6 0Þ;

where e ¼ maxfd1; . . . ; dn;D1; . . . ;Dng. �

4. Local stability analysis

As we have demonstrated in the previous section the integro-differential equation system (3) is equivalent to a system

of ordinary differential-difference equations, therefore the same methods can now be applied as used for the stability

analysis of ordinary differential-difference equations.

e special subcases when mi ¼ 0 and/or li ¼ 0 are obtained by using (10)–(12) and (17)–(19) as appropriate.


Using xiðtÞ and QiðtÞ now to denote deviations of these variables from their equilibrium levels, the linearization of

Eq. (3) reads

_xxiðtÞ ¼ ki ci

Z t

t�di

wðt�

� s; Ti;mi; diÞQiðsÞds�Z t

t�Di

wðt � D; Si; li;DiÞxiðsÞdso; ð21Þ

where wðt � s; T ;mÞ is as defined previously, furthermore ci is the derivative of the best response gi of firm i at theequilibrium. We now state the main result on the eigenvalue structure of the system (21).

Theorem 1. The eigenvalue spectrum of the linearized equation (21) is given by the solutions of the polynomial-exponentialequation

Ynj¼1

ðAjðkÞ � BjðkÞÞ 1

"þXnj¼1

BjðkÞAjðkÞ � BjðkÞ

#¼ 0; ð22Þ

where the quantities AiðkÞ, BiðkÞ are defined by Eqs. (25) and (26) below.

Proof. We look for the solution in the form

xiðtÞ ¼ viekt ði ¼ 1; 2; . . . ; nÞ

as in [1], then Eq. (21) has a similar form as in [2]. In order to obtain the specific expression, we need to compute

integrals of the form:

Imðt; T ; dÞ ¼Z t

t�d

1

C 1m!

mT

� �mþ1


T eks ds ð23Þ

if mP 1, and

I0ðt; T ; dÞ ¼Z t

t�d

1

C 1Te�

t�sT eks ds ð24Þ

if m ¼ 0, where C is given earlier as the normalising constant Cðm; T ; dÞ. From Eq. (23) we have (with new variable

u ¼ t � s)

I0ðt; T ; dÞ ¼Z d

0

1

C 1Te�

uTekðt�uÞ du ¼ 1

CTekt

Z d

0

e�u kþ1Tð Þ du ¼ ekt

CTe�u kþ1

Tð Þ� k þ 1

T

� �" #d

0

¼ ekt

CT ðk þ 1TÞ

1�

� e�d kþ1Tð Þ�

¼ Cð0; T ; dð1þ kT ÞÞCð0; T ; dÞ ð1þ kT Þ�1

ekt:

If mP 1, then by introducing the new variable u ¼ t � s again,

Imðt; T ; dÞ ¼Z d

0

1

C1

m!mT

� �mþ1

ume�muT ekðt�uÞ du ¼

Z d

0

1

C1

m!mT

� �mþ1

ume�muT 1þkT

mð Þ du ekt:

Introduce next the new variable v ¼ u 1þ kTm

� �to obtain

Imðt; T ; dÞ ¼Z d 1þkT

mð Þ

0

1

C1

m!mT

� �mþ1 vm

1þ kTm

� �m e�mvT

dv1þ kT

m

� � ekt ¼C m; T ; d 1þ kT

m

� �� Cðm; T ; dÞ 1

1þ kTm

� �mþ1ekt:

Introduce the following notation:

AiðkÞ ¼k þ ki

C li; Si;Di 1þ kSili

� �Cðli; Si;DiÞ

1þ kSili

��li�1

if li P 1;

k þ kiCð0; Si;Dið1þ kSiÞÞ

Cð0; Si;DiÞð1þ kSiÞ�1

if li ¼ 0

8>>>>>><>>>>>>:

ð25Þ


and

3 In

BiðkÞ ¼�kici

C mi; Ti; di 1þ kTimi

� �� Cðmi; Ti; diÞ

1þ kTimi

��mi�1

if mi P 0;

�kiciCð0; Ti; dið1þ kTiÞÞ

Cð0; Ti; diÞð1þ kTiÞ�1

if mi ¼ 0:

8>>>><>>>>:

ð26Þ

Then the eigenvalue equation of the linearized differential-difference equation system (21) has the form:

AiðkÞvi þ BiðkÞXj 6¼i

vj ¼ 0 ði ¼ 1; 2; . . . ; nÞ ð27Þ

which is equivalent to the determinantal equation

det

A1ðkÞ B1ðkÞ . . . B1ðkÞB2ðkÞ A2ðkÞ . . . B2ðkÞ... ..

. . .. ..

.

BnðkÞ BnðkÞ . . . AnðkÞ

0BBB@

1CCCA ¼ 0: ð28Þ

Notice that this determinant has the same structure as in the case of infinitely continuously distributed lags (see

Chiarella and Szidarovszky, 2000), 3 so Eq. (28) can be rewritten as Eq. (22). �

The main local stability result may now be stated as:

Corollary. The equilibrium of the system (3) is locally asymptotically stable if all roots of Eq. (22) have negative real parts.

Notice that Eq. (22) can be solved by solving nþ 1 equations of more simple structure:

AiðkÞ � BjðkÞ ¼ 0 ðj ¼ 1; 2; . . . ; nÞ ð29Þ

and

1þXnj¼1

BjðkÞAjðkÞ � BjðkÞ

¼ 0: ð30Þ

We will now consider the symmetric case when mi � m, li � l, ki � k, ci � c, Si � S, Ti � T , di � d, and Di � D. Let

q ¼ l if lP 1

1 if l ¼ 0

�and r ¼ m if mP 1;

1 if m ¼ 0:

�

With symmetric initial conditions system (21) becomes one-dimensional and similarly to Eq. (22) we can prove that

in this case Eq. (28) has the special form

k 1

þ kS

q

�lþ1

1

þ kT

r

�mþ1

þ kC l; S;D 1þ kS

q

� �� Cðl; S;DÞ 1

þ kT

r

�mþ1

� kcðn� 1ÞC m; T ; d 1þ kT

r

� �� Cðm; T ; dÞ 1

þ kS

q

�lþ1

¼ 0: ð31Þ

Thus we have the following result.

Theorem 2. The equilibrium of the symmetric dynamic oligopoly with bounded continuously distributed lags is locallyasymptotically stable if the roots of the polynomial-exponential equation (31) have negative real parts.

If both lags d and D are allowed to tend to infinity then the polynomial-exponential equation (31) reduces to the

polynomial equation for the corresponding infinitely continuously distributed lag case in [3].

deed the expressions for AiðkÞ, BiðkÞ reduce to the corresponding ones in [3] when di ! 1 and Di ! 1.


5. The birth of limit cycles

In general equation (31) will have an infinite number of roots, many of which are possibly complex. Thus this class of

dynamic oligopolies is quite prone to exhibiting fluctuating output. In this situation it becomes relevant to consider

whether such fluctuations, could turn into limit cycle motion as certain parameters vary. The aim of this section is to

consider this question.

For the sake of simplicity we restrict our analysis to the symmetric case. It is well-known from bifurcation theory

(see, for example [5]) that limit cycles are possible if there are non-zero pure complex eigenvalues with additional

conditions that will be presented later. Assume therefore that k ¼ ia with some real non-zero a. In order to investigate

Eq. (31) under this situation introduce the following notation:

1

þ iaS

q

�lþ1

¼ AðrÞðaÞ þ iAðiÞðaÞ

with

AðrÞðaÞ ¼Xlþ1

2½ �

v¼0

lþ 1

2v

�a2vS2v

q2vð�1Þv

and

AðiÞðaÞ ¼Xl

2½ �

v¼0

lþ 1

2vþ 1

�a2vþ1S2vþ1

q2vþ1ð�1Þv:

Similarly let

1

þ iaT

r

�mþ1

¼ BðrÞðaÞ þ iBðiÞðaÞ

with

BðrÞðaÞ ¼Xmþ1

2½ �

v¼0

mþ 1

2v

�a2vT 2v

r2vð�1Þv

and

BðiÞðaÞ ¼Xm

2½ �

v¼0

mþ 1

2vþ 1

�a2vþ1T 2vþ1

r2vþ1ð�1Þv:

Notice furthermore that

C l; S;D 1

þ iaS

q

��¼ 1� e�

lD 1þiaSqð Þ

S

Xlk¼0

1

k!

lkDk 1þ iaSq

� �kSk

:

Introduce next the following additional notation:

DðrÞ1 ðaÞ ¼

Xlk¼0

1

k!lkDk

Sk

Xk2½ �

v¼0

k2v

�a2vS2v

q2vð

0@ � 1Þv

1A;

DðiÞ1 ðaÞ ¼

Xlk¼0

1

k!lkDk

Sk

Xk�12½ �

v¼0

k2vþ 1

�a2vþ1S2vþ1

q2vþ1ð

0@ � 1Þv

1A

and notice that

e�lD 1þiaS

qð ÞS ¼ e�

lDS cos

alDq

� i sin

alDq

�:


Similarly,

C m; T ; d 1

þ iaT

r

��¼ 1� e�

md 1þiaTrð Þ

T

Xmk¼0

1

k!mkdk 1þ iaT

r

� �kT k

:

Introduce the further notation,

DðrÞ2 ðaÞ ¼

Xmk¼0

1

k!mkdk

T k

Xk2½ �

v¼0

k2v

�a2vT 2v

r2vð

0@ � 1Þv

1A;

DðiÞ2 ðaÞ ¼

Xmk¼0

1

k!mkdk

T k

Xk�12½ �

v¼0

k2vþ 1

�a2vþ1T 2vþ1

r2vþ1ð

0@ � 1Þv

1A

and notice that

e�md 1þiaT

rð ÞT ¼ e�

mdT cos

amdr

� i sin

amdr

�:

Using the above notation, Eq. (31) with k ¼ ia can be rewritten as

iaðAðrÞðaÞ þ iAðiÞðaÞÞðBðrÞðaÞ þ iBðiÞðaÞÞ þ kCðl; S;DÞ 1

�� e�

lDS cos

alDq

� i sin

alDq

�ðDðrÞ

1 ðaÞ þ iDðiÞ1 ðaÞÞ

ðBðrÞðaÞ

þ iBðiÞðaÞÞ � kcðn� 1ÞCðm; T ; dÞ 1

�� e�

mdT cos

amdr

� i sin

amdr

�ðDðrÞ

2 ðaÞ þ iDðiÞ2 ðaÞÞ

ðAðrÞðaÞ þ iAðiÞðaÞÞ ¼ 0:

Equating the real and imaginary parts to zero we have the following equations:

kcðn� 1Þ ¼ E1ðaÞE2ðaÞ

¼ F1ðaÞF2ðaÞ

ð32Þ

with

E1ðaÞ ¼ �a AðrÞðaÞBðiÞðaÞ�

þ AðiÞðaÞBðrÞðaÞ�þ kCðl; S;DÞ BðrÞðaÞ

�� e�

lDS cos

alDq

DðrÞ1 ðaÞBðrÞðaÞ

�� DðiÞ

1 ðaÞBðiÞðaÞ�

þ sinalDq

ðDðrÞ1 ðaÞBðiÞðaÞ � DðiÞ

1 ðaÞBðrÞðaÞÞ

;

E2ðaÞ ¼1

Cðm; T ; dÞ AðrÞðaÞ�

� e�mdT cos

amdr

ðDðrÞ2 ðaÞAðrÞðaÞ

�� DðiÞ

2 ðaÞAðiÞðaÞÞ

þ sinamdr

ðDðrÞ2 ðaÞAðiÞðaÞ þ DðiÞ

2 ðaÞAðrÞðaÞÞ

;

F1ðaÞ ¼ aðAðrÞðaÞBðrÞðaÞ � AðiÞðaÞBðiÞðaÞÞ þ kCðl; S;DÞ BðiÞðaÞ

�� e�

lDS cos

alDq

ðDðrÞ1 ðaÞBðiÞðaÞ

�þ DðiÞ

1 ðaÞBðrÞðaÞÞ

� sinalDq

ðDðrÞ1 ðaÞBðrÞðaÞ � DðiÞ

1 ðaÞBðiÞðaÞÞ

and

F2ðaÞ ¼1

Cðm; T ; dÞ AðiÞðaÞ�

� e�mdT cos

amdr

ðDðrÞ2 ðaÞAðiÞðaÞ

�þ DðiÞ

2 ðaÞAðrÞðaÞÞ

� sinamdr

ðDðrÞ2 ðaÞAðrÞðaÞ � DðiÞ

2 ðaÞAðiÞðaÞÞ

:

Any real root of Eq. (32) must satisfy

E1ðaÞF2ðaÞ � E2ðaÞF1ðaÞ ¼ 0 ð33Þ


which is a mixed polynomial-trigonometric equation for a. Notice that Eq. (33) might have other roots which are not

solutions of Eq. (32). These additional roots can be identified by simple substitution. Let a be a root of Eq. (32), then

k ¼ ia is an eigenvalue, and there is a functional relation between a and the critical bifurcation value of the parameter

c.Using conditions stated in [5], 4 we may assert that limit cycles are born if Eq. (32) has no other real root a ¼ la

with any integer l for c ¼ c , and if the real part of the derivative dk=dc is non-zero at the root k . The condition on the

other eigenvalues is hard to verify analytically, but computer methods can be used in particular cases.

Notice that

4 Se

dkdc

¼ P1ðkÞP2ðkÞ

; ð34Þ

where

P1ðkÞ ¼ kðn� 1ÞC m; T ; d 1þ kT

r

� �� Cðm; T ; dÞ 1

þ kS

q

�lþ1

and

P2ðkÞ ¼ 1

þ kS

q

�lþ1

1

þ kT

r

�mþ1

þ kðlþ 1Þ Sq

1

þ kS

q

�l

1

þ kT

r

�mþ1

þ kðmþ 1Þ Tr

1

þ kS

q

�lþ1

1

þ kT

r

�m

þ kCðl; S;DÞ

d

dkC l; S;D 1

"þ kS

q

��1

þ kT

r

�mþ1

þ C l; S;D 1

þ kS

q

��ðmþ 1Þ T

r1

þ kT

r

�m#

� kcðn� 1ÞCðm; T ; dÞ

d

dkC m; T ; d 1

"þ kT

r

��1

þ kS

q

�lþ1

þ C m; T ; d 1

þ kT

r

��ðlþ 1Þ S

q1

þ kS

q

�l#:

Hence we may appeal to the Hopf bifurcation theorem [5] to state the following result.

Theorem 3. Let a be a real root of Eq. (32) with c being the derivative of the best response function at the equilibrium,assume there is no other root a ¼ la for any integer l, furthermore at c ¼ c ,

ReP1ðia ÞP2ðia Þ 6¼ 0:

Then there is a limit cycle around the equilibrium.

The Hopf bifurcation theorem does not enable us to say anything about the stability of the limit cycle. To do so in

the context of these dynamic oligopoly models would require specification of the best response function and consid-

eration of conditions arising from normal form theory. However in the present context this may be difficult and nu-

merical simulations are probably the only practical approach. We consider one specific example in Section 8.

6. Pure complex roots in a special case

In this section we consider the special case of S ¼ 0, when no time-lag is assumed in obtaining and implementing

information on the firms� own output. In this case Eq. (31) simplifies to:

ðk þ kÞ 1

þ kT

r

�mþ1

¼ kcðn� 1Þ1� e�

md 1þkTrð Þ

TPm

k¼0mkdk

k!T k 1þ kTr

� �k1� e�

mdTPm

k¼0mkdk

k!T k

: ð35Þ

Consider first the case of m ¼ 0. Then the above equation is quadratic:

ðk þ kÞð1þ kT Þ ¼ kcðn� 1Þ:

e Section 11.1.


If k ¼ ia, then this equation becomes

ðia þ kÞð1þ iaT Þ ¼ kcðn� 1Þ:

Equating the real and imaginary parts we have

k � a2T ¼ kcðn� 1Þ and að1þ kT Þ ¼ 0:

Therefore a ¼ 0 and c ¼ 1=ðn� 1Þ is the only solution, which implies that the birth of limit cycles is not guaranteed

in this case given that c < 0, and the pure complex root is zero.

Consider next the case of m ¼ 1. Then Eq. (35) reduces to

ðk þ kÞð1þ kT Þ2 ¼ kcðn� 1Þ1� e�

dð1þkT ÞT 1þ d

T ð1þ kT Þ� �

1� e�dT 1þ d

T

� � : ð36Þ

If k ¼ ia, then the left hand side of (36) is

ðia þ kÞð1þ 2iaT � a2T 2Þ ¼ ð�a22T þ k � ka2T 2Þ þ iða � a3T 2 þ 2kaT Þ:

Similarly the right hand side is

kcðn� 1Þ1� e�

dT 1þ d

T

� � 1

�� e�

dT ðcos da � i sindaÞ 1

þ dTþ dai

�

¼ kcðn� 1Þ1� e�

dT 1þ d

T

� � 1

�� e�

dT cos da 1

�þ dT

�þ sin daðdaÞ

þ ie�

dT

�� cos daðdaÞ þ sin da 1

þ dT

� :

Comparing the real and imaginary parts we obtain

kcðn� 1Þ ¼ð�a22T þ k � ka2T 2Þ 1� e�

dT 1þ d

T

� �� 1� e�

dT cos da 1þ d

T

� �þ sin daðdaÞ

! " ¼ða � a3T 2 þ 2kaT Þ 1� e�

dT 1þ d

T

� �� e�

dT � cos daðdaÞ þ sin da 1þ d

T

� �! " : ð37Þ

This equation can be rewritten as a single polynomial-trigonometric equation for a, viz.,

GðaÞ � ½k � a2ð2T þ kT 2Þ�e�dT

�� da cos da þ 1

þ dT

�sin da

� ½að1þ 2kT Þ � a3T 2� 1

�� e�

dT 1

�þ dT

�cos da þ da sin da

¼ 0: ð38Þ

All eigenvalues satisfy Eq. (38). However there might be additional roots of (38) which are not eigenvalues. They can

be identified by simple substitution into Eq. (36). For instance, a ¼ 0 is always a root, but k ¼ 0 solves Eq. (36) for only

c ¼ 1=ðn� 1Þ which is not possible, since c < 0.

We need at least one non-zero real root in order to establish the birth of limit cycles. Let a be a non-zero real root,

then the corresponding value c of the bifurcation parameter is obtained from Eq. (37). The derivative dk=dc (evaluatedat the equilibrium) can also be obtained from Eq. (36) as

dkdc

¼ P1ðkÞP2ðkÞ

ð39Þ

with

P1ðkÞ ¼ 1

�� e�

dð1þkT ÞT 1

þ dTð1þ kT Þ

�kðn� 1Þ

and

P2ðkÞ ¼ 1

�� e�

dT 1

þ dT

�½ð1þ kT Þ2 þ ðk þ kÞ2T ð1þ kT Þ� � kcðn� 1Þ d

2

Te�

dð1þkT ÞT ð1þ kT Þ:

If Eq. (36) has no other root a ¼ la with any integer l for c ¼ c , and if the real part of this derivative at a ¼ a and

c ¼ c is non-zero, then there are limit cycles around the equilibrium.


Note that by letting d ! 1 in Eq. (38) we obtain

a 2 ¼ 2kT þ 1

T 2ð40Þ

and then from the first equality in (37)

c ¼ � 2

ðn� 1Þ1

kT

�þ kT þ 2

; ð41Þ

these being indeed the results obtained for this case by Chiarella and Szidarovszky [3]. In the bounded lag case Eq. (38)

possibly has an infinite number of roots and hence, an infinite number of bifurcation values c are possible. In order to

appreciate the much richer set of possibilities it is necessary to consider a numerical example.

7. Numerical example

We consider from Section 6 the special case of m ¼ 1 with T ¼ 1, k ¼ 3 and n ¼ 10.

The key to understanding the overall structure of the eigenvalues and the critical values of c is to consider how the

roots of Eq. (38) evolve as the parameter d increases from 0 through 5 (the value that for our purposes may be regarded

as infinity).

Since we are only concerned with non-zero roots the ensuing discussion ignores the root at a ¼ 0. We note that Eq.

(38) is symmetric with respect to a, so we only consider positive values. Fig. 3a shows a plot of GðaÞ at d ¼ 2:5 for a up

to 30 illustrating the cyclically increasing nature of this function. Some elementary manipulations on GðaÞ for large aindicate easily that GðaÞ must have an infinite number of real roots. A root a 6¼ 0 may lead to the birth of limit cycles if

none of the other roots is integer multiple of a and gives the same critical value of c.Fig. 3b shows the more detailed structure of GðaÞ for a up to about 7.5, for three separate values of d. At d ¼ 3:0 (the

blue curve) we can view the intersections with the horizontal axis resulting in the first seven roots. At d ¼ 3:17 (the red

curve) we see how the first two roots disappear via a tangency, the third root declines, roots four and five decline and

move closer together, and roots six and seven appear. At d ¼ 3:40 (the black curve) the third root continuous to decline,

the fourth and fifth roots are close to disappearing and the sixth and seventh roots are just disappearing. For this set of

parameters this is the only ‘‘reversal’’ in the order in which roots disappear that we have observed.

The foregoing observations on Fig. 3b should then make fairly transparent how Fig. 4a is constructed. The blue

‘‘tongue’’ shows the first (lower) and second (upper) roots which come together and then disappear at d ¼ 3:17. Theblack curve show the third root declining continuously to its limiting value at d ¼ 5, this limiting value is in fact close toffiffiffi7

pas would be expected from Eq. (40). The red tongue shows the fourth (lower) and fifth (upper) roots which came

together and then disappear at d ¼ 3:48. The green tongue shows the sixth and seventh roots which come together and

then disappear at d ¼ 3:41. The tongues for roots eight, nine and ten, eleven are also shown in Fig. 4a. The full range of

tongues up to a ¼ 25 are shown in Fig. 4b, this is close to the largest value of a that is relevant given that d ¼ 5 is

considered to be infinity for the purposes of these illustrative calculations.

We next use the values of a in Fig. 4a to construct, via Eq. (37), the corresponding critical values of c at which a pair

of pure complex roots occurs. These are shown in Fig. 5a with the colours corresponding to the tongues (except in the

case of root three) shown in Fig. 4a. The limiting value to which the black curve (corresponding to the third root for a)tends is in fact close to )32/27 as expected from Eq. (41). Fig. 5b shows the whole sequence of critical c curves cor-

responding to the sequence of tongues in Fig. 4b.

In order to obtain a more detailed global picture of the dynamics it would be necessary to specify functional forms

for the reaction function g and calculate for example a bifurcation diagram with respect to d. This we do for one specific

example in the next section.

8. A specific reaction function

In order to generate a specific example we assume linear cost functions and the hyperbolic inverse demand function

f ðQÞ ¼ A=Q with some A > 0. The profit of firm j is then given as

pj ¼ xjA

xj þ Qj� cjxj;

Fig. 4. The relation between the roots of Eq. (38) and the lag length.

0 10 20 30α

-2E+5

-1E+5

0E+0

1E+5

2E+5

k = 3 δ = 2.5T = 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0α

0

40

80

k = 3

δ= 3.0 3.17 3.40

T = 1

(a)

(b)

Fig. 3. The roots of Eq. (38).


1 2 3 4 5δ

-200

-160

-120

-80

-40

0

2 4-20

-10

0

γ∗

δ

γ∗

(a) (b)

Fig. 5. The relation between the emergence of pure complex roots and the lag length.


(where cj is the marginal cost) and the best response of this firm is

gjðQjÞ ¼0; if Qj PA=cj;ffiffiffiffiffiffiffiffiAQj

cj

s� Qj; otherwise:

8><>: ð42Þ

We continue to focus on the symmetric case considered in Section 6 where in addition we have set cj ¼ c. At a

symmetric equilibrium Q j ¼ ðn� 1Þx and Q ¼ nx , and x is the solution of equation

x ¼ gððn� 1ÞxÞ;

which by use of (42) yields

x ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðn� 1Þx

c

r � ðn� 1Þx

!;

implying that

x ¼ Aðn� 1Þcn2

:

It is easy to see that for n > 2, the function gj is strictly decreasing in Qj in the neighborhood of the equilibrium,

hence satisfying the condition on the best response function made in Section 3.

From Eqs. (8) and (9) it is easy to verify that

qðkÞ i;m ¼ ðn� 1Þx 1

� e

mdT

Xkj¼1

mdT

� �jj!

!

for k ¼ 0; 1; . . . ;m, so in the special case of k ¼ m,

qðmÞ i;m ¼ ðn� 1Þx Cðm; T ; dÞ

implying that

Qe i ¼ ðn� 1Þx :

Using the reaction function (42) we have simulated the system of Lemma 2 for the case S ¼ 0 (as in Section 6) and

k ¼ 3, n ¼ 10 and m ¼ 1 (as in Section 7) and A=c ¼ 1. The lag d was taken as a bifurcation parameter and the bi-

furcation diagram in Fig. 6 was calculated for T ¼ 1 and T ¼ 1:5 over the range 0 < d6 5. Recall that from a practical

point of view d ¼ 5 may be regarded as the infinitely distributed lag case. We see that compared to this limiting case the

bounded lag weighting schemes increase the likelihood of limit cycle motions, depending on the value of T . This one

0 1 2 3 4 5δ

0.00

0.05

0.10

0.15

0.20

0.25

q

n = 10, K = 3, T = 1 & 1.5

Fig. 6. Bifurcation diagram of output with respect to lag length.


numerical example illustrates how the bounded lag weighting scheme can more readily yield fluctuating dynamics. We

leave for future research a more extensive numerical study of this model, which should study other reaction functions

and allow the firms to differ in various ways.

9. Conclusions

In this paper we have introduced a weighting function that caters for bounded continuously distributed lags in

dynamic economic models. In this way past data are averaged only over a bounded interval thus avoiding the use of

very old or less relevant economic data. We have used this weighting function to capture the effects of information and

implementation lags about firms� own, as well as rivals�, output in a fairly general class of dynamic oligopoly models.

We have formulated the dynamics of the resulting oligopoly as a system of integro-differential equations and shown

how this may be reduced to a system of ordinary differential-difference equations. We have then employed standard

techniques from the theory of differential-difference equations to analyse the local stability of the equilibrium. We have

then applied the Hopf-bifurcation theorem to determine conditions on the model parameters under which limit cycle

motion may be born (or destroyed). We have considered in some detail the special case of a symmetric oligopoly and

examined it numerically for a particular set of parameters. This example indicates how the dynamic structure becomes

much more complex compared to the corresponding (and limiting) infinitely continuously distributed lag case. In that

case there is just one critical value of which limit cycles are born (or destroyed), whereas in the bounded lag case there

may be many such values. Dynamic oligopoly models with bounded continuously distributed lags are thus far more

easily able to exhibit limit cycle and possibly more complex motion. We have given one simple numerical example that

seems to confirm these insights from the theoretical discussion.

We believe we have taken an analytical investigation of the stability of this class of dynamic oligopoly models as far

as is possible. It would be of interest to complement this study with a more extensive simulation study to see how Figs.

3–6 are altered by allowing the firms to differ in some way, also by considering a higher value of the parameter m in

Section 6. It would also be of interest to analyse globally the types of attractors, as well as their basins of attraction, that

may occur. Whilst in this paper we have only considered bounded continuously distributed lags within dynamic oli-

gopoly models, we believe their implications within a range of other dynamic economic models should be examined.

Such lags surely reflect more accurately the reality of economic decision making and it seems difficult to believe that the

stability structure displayed by the example in Sections 7 and 8 is confined merely to dynamic oligopoly models.

Acknowledgements

The authors are indebted to Dr. Peiyuan Zhu for writing the computer programs that generated the diagrams. The

authors also with to thank Professor Laura Gardini for a number of very insightful comments that helped to clarify

some misconceptions in an earlier version of this paper.


References

[1] Bellman R, Cooke KL. Differential-difference equations. New York: Academic Press; 1963.

[2] Chiarella C, Khomin A. An analysis of the complex dynamic behaviour of nonlinear oligopoly models with time delays. Chaos,

Solitons & Fractals 1996;7(2):2049–65.

[3] Chiarella C, Szidarovszky F. Limit cycles in nonlinear oligopolies with continuously distributed time delays. In: Dror M, editor.

Modelling uncertainty. Dordrecht: Kluwer Academic Publishers; 2002. p. 249–68.

[4] Cushing JM. Integro-differential equations and delay models in population dynamics. Berlin: Springer-Verlag; 1977.

[5] Hale JK, Verduyn Lunel SM. Introduction to functional differential equations. Berlin: Springer-Verlag; 1993.

[6] Invernizzi S, Medio A. On lags and chaos in economic models. J Math Econ 1991;20:512–50.

[7] Okuguchi K. Expectations and stability in oligopoly models. Berlin: Springer-Verlag; 1976.

[8] Okuguchi K, Szidarovszky F. The theory of oligopoly with multi-product firms. 2nd ed. Berlin: Springer-Verlag; 1999.

[10] Russel AM, Rickard J, Howroyd TD. The effects of delays on the stability and rate of convergence to equilibrium of oligopolies.

Econ Rec 1986;62:194–8.

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Bounded continuously distributed delays in dynamic oligopolies